List of practice Questions

A compound microscope has an objective of focal length $1.25\,\mathrm{cm}$ and an eyepiece of focal length $5\,\mathrm{cm}$ separated by a distance of $7.5\,\mathrm{cm}$. The total magnification produced by the microscope when the final image forms at infinity is

The property of light that explains the formation of coloured images due to thick lenses is

The ratio of the time periods of a simple pendulum at heights $2R_E$ and $3R_E$ from the surface of the earth is ($R_E$ is radius of the earth)

Two wires A and B made of same material and areas of cross-section in the ratio $1:2$ are stretched by same force. If the masses of the wires A and B are in the ratio $2:3$, then the ratio of the elongations of the wires A and B is

Water is filled in a tank up to a height of 20 cm from the bottom of the tank. Water flows through a hole of area $1\text{ mm}^2$ at its bottom. The mass of the water coming out from the hole in a time of 0.6 s is
(Density of water $= 1000\text{ kg m}^{-3}$ and acceleration due to gravity $= 10\text{ ms}^{-2}$)

For which of the following Reynold's number, a flow is streamlined?

If the system of blocks shown in the figure is released from rest, the ratio of the tensions $T_1$ and $T_2$ is (Neglect the mass of the string shown in the figure)

If the component of the vector $\vec{A}$ along the vector $\vec{B}$ is twice the component of $\vec{B}$ along $\vec{A}$, then the ratio of magnitudes of vectors $\vec{A}$ and $\vec{B}$ is

A body projected vertically up with an initial speed of $10\text{ ms}^{-1}$ reaches the point of projection after sometime with a speed of $8\text{ ms}^{-1}$. The maximum height reached by the body is (Acceleration due to gravity $= 10\text{ ms}^{-2}$)

Due to global warming, if the ice in the polar region melts and some of this water flows to the equatorial region, then

Bose-Einstein statistics is applicable to particles with

If L and C are inductance and capacitance respectively, then the dimensional formula of $(LC)^{-\frac{1}{2}}$ is

Two bodies are projected from the same point with the same initial velocity 'u' making angles '$\theta$' and ($90^\circ-\theta$) with the horizontal in opposite directions. The horizontal distance between their positions when the bodies are at their maximum heights is

If \(I_n = \int \frac{1}{(x^2+1)^n} dx\), then \(2n I_{n+1} - (2n-1) I_n =\)

\(\int \frac{x^3}{x^4 + 3x^2 + 2} dx =\)

If \(\int \frac{dx}{(x^2+9)\sqrt{x^2+16}} = \frac{1}{3\sqrt{7}} \tan^{-1} \left( K \frac{x}{\sqrt{16+x^2}} \right) + c\), then \(K =\)

\(\lim_{n \to \infty} \frac{1}{n^2} \left[ e^{1/n} + 2e^{2/n} + 3e^{3/n} + \dots + 2n e^{2n/n} \right] =\)

Let m, n, p, q be four positive integers. If \(\int_0^{2\pi} \sin^m x \cos^n x dx = 4 \int_0^{\pi/2} \sin^m x \cos^n x dx\), \(\int_0^{2\pi} \sin^p x \cos^q x dx = 0\), \(a = m+n+p\) and \(b = m+n+q\), then

The area of the region bounded by the curves \(y=x^3\), \(y=x^2\) and the lines \(x=0\) and \(x=2\) is

If \(y = (\sin^{-1}x)^2\), then \((1-x^2)\frac{d^2y}{dx^2} - x\frac{dy}{dx} =\)

The radius of a cone of height 9 units is changed from 2 units to 2.12 units. The exact change and approximate change in the volume of the cone are respectively

The local maximum value \(l\) and local minimum value \(m\) of \(f(x) = \frac{x^2+2x+2}{x+1}\) in \(\mathbb{R} - \{-1\}\) exist at \(\alpha, \beta\) respectively, then \(\frac{l+m}{\alpha+\beta} =\)

\(P(5,2)\) is a point on the curve \(y=f(x)\) and \(\frac{7}{2}\) is the slope of the tangent to the curve at P. The area of the triangle formed by the tangent and the normal to the curve at P with x-axis is

If a particle is moving in a straight line so that after \(t\) seconds its distance \(S\) (in cms) from a fixed point on the line is given by \(S = f(t) = t^3 - 5t^2 + 8t\) then the acceleration of the particle at \(t=5\) sec is (in cm/sec\(^2\))

If \(f:[a,b] \to [c,d]\) is a continuous and strictly increasing function, then \(\frac{d-c}{b-a}\) is